Lecture 2 Goals: (Highlights of Chaps. 1 & 2.1-2.4) • Conduct order of magnitude calculations, • Determine units, scales, significant digits (in discussion or on your own) • Distinguish between Position & Displacement • Define Velocity (Average and Instantaneous), Speed • Define Acceleration • Understand algebraically, through vectors, and graphically the relationships between position, velocity and acceleration • Perform Dimensional Analysis
Reading Quiz Displacement, position, velocity & acceleration are the main quantities that we will discuss today. Which of these 4 quantities have the same units • Velocity & position • Velocity & acceleration • Acceleration & displacement • Position & displacement • Position & acceleration
Perspective Length/Time/Mass Distance Length (m) Radius of Visible Universe 1 x 1026 To Andromeda Galaxy 2 x 1022 To nearest star 4 x 1016 Earth to Sun 1.5 x 1011 Radius of Earth 6.4 x 106 Sears Tower 4.5 x 102 Football Field 1 x 102 Tall person 2 x 100 Thickness of paper 1 x 10-4 Wavelength of blue light 4 x 10-7 Diameter of hydrogen atom 1 x 10-10 Diameter of proton1 x 10-15 Universal standard: The speed of light is defined to be exactly 299 792 458 m/s and so one measures how far light travels in 1/299 792 458 of a second
Time IntervalTime (s) Age of Universe 5 x 1017 Age of Grand Canyon 3 x 1014 Avg age of college student 6.3 x 108 One year3.2 x 107 One hour3.6 x 103 Light travel from Earth to Moon 1.3 x 100 One cycle of guitar A string 2 x 10-3 One cycle of FM radio wave 6 x 10-8 One cycle of visible light 1 x 10-15 Time for light to cross a proton 1 x 10-24 World’s most accurate timepiece: Cesium fountain Atomic Clock Lose or gain one second in some 138 million years
Order of Magnitude Calculations / EstimatesQuestion: If you were to eat one french fry per second, estimate how many years would it take you to eat a linear chain of trans-fat free french fries, placed end to end, that reach from the Earth to the moon? • Need to know something from your experience: • Average length of french fry: 3 inches or 8 cm, 0.08 m • Earth to moon distance: 250,000 miles • In meters: 1.6 x 2.5 X 105 km = 4 X 108 m • 1 yr x 365 d/yr x 24 hr/d x 60 min/hr x 60 s/min = 3 x 107 sec
Dimensional Analysis (reality check) • This is a very important tool to check your work • Provides a reality check (if dimensional analysis fails then there is no sense in putting in numbers) • Example When working a problem you get an expression for distance d = v t 2( velocity · time2 ) Quantity on left side d L length (also T time and v m/s L / T) Quantity on right side = L / T x T2 = L x T • Left units and right units don’t match, so answer is nonsense
Exercise 1Dimensional Analysis • The force (F) to keep an object moving in a circle can be described in terms of: • velocity (v, dimension L / T) of the object • mass (m, dimension M) • radius of the circle (R, dimension L) Which of the following formulas for Fcould be correct ? Note: Force has dimensions of ML/T2 orkg-m / s2 F = mvR (a) (c) (b)
F = mvR Exercise 1Dimensional AnalysisWhich of the following formulas for Fcould be correct ? • • • Note: Force has dimensions of ML/T2 Velocity (n, dimension L / T) Mass (m, dimension M) Radius of the circle (R, dimension L)
Tracking changes in position: VECTORS • Position • Displacement (change in position) • Velocity (change in position with time) • Acceleration (change in velocity with time) • Jerk (change in acceleration with time) Be careful, only vectors with identical units can be added/subtracted
10 meters Joe +x -x 10 meters O Motion in One-Dimension (Kinematics) Position • Position is usually measured and referenced to an origin: • At time= 0 seconds Pat is 10 meters to the right of the lamp • Origin lamp • Positive direction to the right of the lamp • Position vector :
10 meters 15 meters xi Pat Displacement • One second later Pat is 15 meters to the right of the lamp • Displacement is just change in position • x = xf - xi Δx xf O xf = xi + x x = xf - xi = 5 metersto the right ! t = tf - ti= 1 second Relating x to t yields velocity
Average VelocityChanges in positionvsChanges in time • Average velocity = net distance covered per total time, includes BOTH magnitude and direction • Pat’s average velocity was 5 m/s to the right
Average Speed • Speed, s, is usually just the magnitude of velocity. • The “how fast” without the direction. • However: Average speed references the total distance travelled • Pat’s average speed was 5 m/s
Exercise 2Average Velocity x (meters) 6 4 2 0 t (seconds) 1 2 3 4 -2 What is the magnitude of the average velocity over the first 4 seconds ? (A) -1 m/s (B) 4 m/s (C) 1 m/s (D) not enough information to decide.
Average Velocity Exercise 3What is the average velocity in the last second (t = 3 to 4) ? • 2 m/s • 4 m/s • 1 m/s • 0 m/s x (meters) 6 4 2 -2 t (seconds) 1 2 3 4
x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 turning point Average Speed Exercise 4What is the average speed over the first 4 seconds ?0 m to -2 m to 0 m to 4 m 8 meters total • 2 m/s • 4 m/s • 1 m/s • 0 m/s
x t Instantaneous velocity • Instantaneous velocity, velocity at a given instant • If a position vs. time curve then just the slope at a point
x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 Exercise 5Instantaneous Velocity • Instantaneous velocity, velocity at a given instant What is the instantaneous velocity at the fourth second? (A) 4 m/s (B) 0 m/s (C) 1 m/s (D) not enough information to decide.
Exercise 5Instantaneous Velocity x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 What is the instantaneous velocity at the fourth second? (B) 0 m/s (A) 4 m/s (C) 1 m/s (D) not enough information to decide.
x t vx t Key point: position velocity • (1) If the position x(t)is known as a function of time, then we can find instantaneous or average velocity v • (2) “Area” under the v(t) curve yields the displacement • A special case: If the velocity is a constant, then x(Δt)=v Δt + x0
v0 v1 v1 v1 v3 v5 v2 v4 v0 Acceleration • A particle’s motion often involve acceleration • The magnitude of the velocity vector may change • The direction of the velocity vector may change (true even if the magnitude remains constant) • Both may change simultaneously • Now a “particle” with smoothly decreasing speed
v1 v0 v3 v5 v2 v4 a a a a a a a t t 0 Constant Acceleration • Changes in a particle’s motion often involve acceleration • The magnitude of the velocity vector may change • A particle with smoothly decreasing speed: Dt v(Dt)=v0 + a Dt v a Dt = area under curve = Dv (an integral)
Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero • Position, velocity & acceleration are all vectors and must be kept separate (i.e., they cannot be added to one another)
Assignment • Reading for Tuesday’s class • Finish Chapter 2 & all of 3 (vectors) • HW 0 due soon, HW1 is available